Need a way to do a "loop cut" on a triangle fan
One way is to use vertex bevel (&#9096; Ctrl&#8679; ShiftB): 



Once the triangle fan is converted to quads you can add loopcuts. In the gif I converted the ngon resulting from the bevel to a triangle fan with Poke (&#9095; AltP).

As an alternative to bevel, you might also try Inset (I):